L(s) = 1 | + 2.44·3-s + 2.44i·5-s + (2.44 + i)7-s + 2.99·9-s + 2i·11-s + 2.44i·13-s + 5.99i·15-s − 4.89i·17-s − 2.44·19-s + (5.99 + 2.44i)21-s + 4i·23-s − 0.999·25-s + 4·29-s + 4.89·31-s + 4.89i·33-s + ⋯ |
L(s) = 1 | + 1.41·3-s + 1.09i·5-s + (0.925 + 0.377i)7-s + 0.999·9-s + 0.603i·11-s + 0.679i·13-s + 1.54i·15-s − 1.18i·17-s − 0.561·19-s + (1.30 + 0.534i)21-s + 0.834i·23-s − 0.199·25-s + 0.742·29-s + 0.879·31-s + 0.852i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.377 - 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.377 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.084632263\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.084632263\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (-2.44 - i)T \) |
good | 3 | \( 1 - 2.44T + 3T^{2} \) |
| 5 | \( 1 - 2.44iT - 5T^{2} \) |
| 11 | \( 1 - 2iT - 11T^{2} \) |
| 13 | \( 1 - 2.44iT - 13T^{2} \) |
| 17 | \( 1 + 4.89iT - 17T^{2} \) |
| 19 | \( 1 + 2.44T + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 - 4T + 29T^{2} \) |
| 31 | \( 1 - 4.89T + 31T^{2} \) |
| 37 | \( 1 + 8T + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 6iT - 43T^{2} \) |
| 47 | \( 1 + 4.89T + 47T^{2} \) |
| 53 | \( 1 - 4T + 53T^{2} \) |
| 59 | \( 1 - 2.44T + 59T^{2} \) |
| 61 | \( 1 - 7.34iT - 61T^{2} \) |
| 67 | \( 1 - 2iT - 67T^{2} \) |
| 71 | \( 1 + 10iT - 71T^{2} \) |
| 73 | \( 1 + 14.6iT - 73T^{2} \) |
| 79 | \( 1 + 6iT - 79T^{2} \) |
| 83 | \( 1 - 2.44T + 83T^{2} \) |
| 89 | \( 1 - 14.6iT - 89T^{2} \) |
| 97 | \( 1 + 4.89iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.244372137528624851637911493154, −8.691309373609650826825604557208, −7.85151292590505657359425979366, −7.22217983030891246768656670719, −6.55745040968756918882947676452, −5.21434628590146253874550988409, −4.32090201298362643699747114542, −3.28809623886295478360444267694, −2.51231382936894167438888931944, −1.79214795135285516117430169771,
1.01050733886054927448279798545, 2.04039073266894559225728347601, 3.15454358769357933207399878380, 4.12277068882847408310164497820, 4.80363124088719428589376227549, 5.81347232657177139814617120271, 6.95530542648761686481935987406, 8.116554340662671246009462246981, 8.422810686289413578286156060068, 8.625900788160267614341615553117